What do you mean by "Paris"? If it's the City of Paris (Paris intra-muros), then it's not comparable to London in terms of size or density. IMO, for the purposes of this discussion, Paris should mean the whole Paris region, since most of the people live outside the actual city limits. And in those areas, access to public transportation is hit or miss. Some people are close to suburban trains, but many are not.
Then, another consideration, which is also very important, is what the available transportation actually looks like. By that I mean how often are there trains, how reliable are they? And, in Paris and probably Central London, too, are you actually able to get on board, or do you need to wait 3 trains packed to the brim?
I don't know about London, but in Paris, the suburban trains have quite poor punctuality.
Note that most car traffic in Paris is actually people from outside the city proper, so those who are most affected by these transit issues.
Additionally, a lot of traffic also goes from suburb to suburb, which, currently, is a terribly bad joke transport-wise. When I was in college, the drive from my parents' house was around 20-30 minutes. Public transit was over one hour with multiple changes, one of which had around one minute of leeway before a 30 minute wait. They are building new circular lines around Paris, but they won't be ready for a few years.
As someone who ever only walks or takes public transit I'm all for limiting car noise and pollution. But what I'd love to see is some form of improvement of the offer (a carrot). Riding around packed like sardines in trains with questionable reliability is a tough sell. I'm lucky enough I can modulate my commute hours to avoid peak times, but not everyone is so lucky. Right now, the city is mostly spending money on making driving hell (all stick).
And bikes are fine, I guess, if you have where to store them. I wouldn't leave any kind of bike unattended around my office. There's also a bike sharing scheme which used to be nice, but for a few months now it's basically impossible to find a usable bike. And I tend to avoid peak times for those, too.